退化时滞微分系统的周期解及稳定性
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摘要
在实际系统中,退化、时滞现象总是普遍存在的,例如控制系统、金融系统、化工系统、工业工程系统、生态系统、电力系统等,稳定性理论是退化时滞微分系统的一个基础结构特征,自从俄国数学力学家Lyapunov建立了常微分方程稳定性理论以来,很多学者在此基础上研究了退化时滞微分系统的稳定性理论,极大地推动了稳定性理论和方法的不断发展和创新,已经取得了大量的研究成果.同时在实际系统中,周期现象也经常出现,因而周期问题也是一个重要的研究方向.
本文对退化时滞微分系统的周期解及稳定性进行了研究并得出了一些结论.全文共分为如下四章.
第一章主要介绍问题的一些背景知识及本文所做的主要工作,给出本文所需的一些预备知识.
第二章研究了如下退化中立型系统的周期解问题.
第三章研究了n维退化时滞微分系统Ex(t)=Ax(t)+Bx(t-τ)全时滞稳定的代数判据.
第四章研究了一类具有分布时滞的Lurie直接控制系统：和一类具有分布时滞的Lurie间接控制系统：的绝对稳定性.
There were many degenerate and time delay phenomena existing in many practi-cal fields, such as control system,financial system,chemical system,industrial project sys-tem,ecosystem,power system and so on.Stability is a basic structure characteristic of de-generate differential systems with delays.Since the Russian mathematician mechanics Lya-punov found stability theory of ordinary differential equations,many scholars have studied the stability theory of degenerate differential systems with delays,which promoted the con-tinuous development and innovation of degenerate differential systems and has obtained a lot of research achievement.At the same time, periodic phenomena often occur in practical fields, and thus the periodic problem is also an important aspect of the study.
This paper deals with periodic solution and stability of degenerate differential systems with delays, many important results are also given in it. There are four chapters in this paper.
In chapter1, some background knowledge of degenerate differential systems with delays is introduced, and the preliminary knowledge which is necessary in the paper is given.
In chapter2, we consider the periodic solution of the following system:
In chapter3,we investigate the stability of the following system: Ex(t)=Ax(t)+Bx(t-τ)
In chapter4, we study the stability of the following two systems: and
本文对退化时滞微分系统的周期解及稳定性进行了研究并得出了一些结论.全文共分为如下四章.
第一章主要介绍问题的一些背景知识及本文所做的主要工作,给出本文所需的一些预备知识.
第二章研究了如下退化中立型系统的周期解问题.
第三章研究了n维退化时滞微分系统Ex(t)=Ax(t)+Bx(t-τ)全时滞稳定的代数判据.
第四章研究了一类具有分布时滞的Lurie直接控制系统：和一类具有分布时滞的Lurie间接控制系统：的绝对稳定性.
There were many degenerate and time delay phenomena existing in many practi-cal fields, such as control system,financial system,chemical system,industrial project sys-tem,ecosystem,power system and so on.Stability is a basic structure characteristic of de-generate differential systems with delays.Since the Russian mathematician mechanics Lya-punov found stability theory of ordinary differential equations,many scholars have studied the stability theory of degenerate differential systems with delays,which promoted the con-tinuous development and innovation of degenerate differential systems and has obtained a lot of research achievement.At the same time, periodic phenomena often occur in practical fields, and thus the periodic problem is also an important aspect of the study.
This paper deals with periodic solution and stability of degenerate differential systems with delays, many important results are also given in it. There are four chapters in this paper.
In chapter1, some background knowledge of degenerate differential systems with delays is introduced, and the preliminary knowledge which is necessary in the paper is given.
In chapter2, we consider the periodic solution of the following system:
In chapter3,we investigate the stability of the following system: Ex(t)=Ax(t)+Bx(t-τ)
In chapter4, we study the stability of the following two systems: and
引文
[1]蒋威.退化时滞微分系统[M].合肥：安徽大学出版社,1998.
[2]郑祖庥.泛函微分方程[M].合肥：安徽教育出版社,1992.
[3]李森林,温立志.泛函微分方程[M].长沙：湖南科学技术出版社,1982.
[4]谢胜利.线性定常中立型大系统的稳定性[J].控制理论与应用,1989,6(3)：7-78.
[5]S.L.Campbell.Singular Systems of Differential Equation(II),Pitman,1982.
[6]L.Dai. Singular Control Systems[M]. Berlin, New York, Springer-Verlag,1989.
[7]廖晓昕.稳定性的理论、方法和应用[M].武汉：华中科技大学出版社,2002.
[8]Jack K Hale & Sjoerd M Verduyn Lunel. Introduction to Functional Differential Equations[M]. Berlin, New York:Springer-Verlay,1992.
[9]蒋威,郑祖麻.退化中立型微分系统的常数变易公式和通解[J].应用数学学报,1998,21(4)：562-570.
[10]周宗福.一般退化时滞微分系统解的存在性及通解[J].数学研究,1998,31(4)：411-416.
[11]Hale J.Theory of functional differential equations[M].New York:Springer-Verlag,1997.
[12]郑大钟.线性系统理论[M].北京：清华大学出版社,1998.
[13]Jiang Wei. On the solvability of singular differential delay systems with variable coefficients[J]. Int.J.Dynamical Systems and Differential Equations,2008,1 (4):245-249.
[14]张海,蒋威.一般退化中立型微分系统解的存在性及通解[J].合肥工业大学学报,2007,30(5)：63-633.
[15]时宝,张德存,盖明久.微分方程理论及其应用[M].北京：国防工业出版社,2005.
[16]Jiang Wei, Zheng Zuxiu.On the degenerate differential systems with delay[J].Ann of Diff. Eqs., 1998,14(2):204-211.
[17]H.H.Rosenbrock. Structural properties of linear dynamical systems.Int.J.Contr.20(2) (1974) 191-202.
[18]Jiang Wei. Function-Controllability of Nonlinear Singular Delay Differential Control System-s[J]. Acta Mathematica Sinica,2006,49(5):1153—1162.
[19]D.G.Luenberger.Dynamical equation in descriptor form,IEEE Trans.Autom.Contr. 22(3)(1977)312-321.
[20]J.D.Cobb.Controllability,observability and duality in singular systems,IEEE Tran-s.Autom.Contr.29(1984)1076-1082.
[21]P.Kunkel,V.Mehrmann.Differential-algebraic equations.European Mathematical Society,2006.
[22]梁家荣,刘永清.广义系统的周期解,控制理论与应用.Vol.14,No.4,1997,595-598.
[23]Andras Varga,Paul,Van,Doorren.Computing the zeros of periodic descriptor systems.Systems and Control Letter.Vol.50,2003,371-381.
[24]周宗福,李蕾,王敬丰,胡秀林.一类退化中立型微分系统的周期解[J].数学物理学报,2006,26A(7)：1025-1030.
[25]Zhou Guifang.Zhang Hai,Jiang Wei.Periodic Solutions of Singular Neutral Differential Systems with Infinite Delay[J].Chin.Quart.J.of Math.2011,26(1):56-60.
[26]蒋威.退化时滞微分方程的周期解问题[J].应用数学学报,2003,26(2)：280-285.
[27]崔冬玲.含有两个时滞项的退化时滞微分方程周期解[J].生物数学学报,2010,25(2)：241-248.
[28]张作元.滞后性方程x(t)=Ax(t)+Bx(x-τ)全时滞稳定的代数判据[J].科学通报,1986：1768-1771.
[29]Lefsches S.Stability of nonlinear control systems[M].New York:Academic Press,1965.
[30]Liao X X,Yu P.Sufficient and necessary conditions for the absolute stability of time delay Lurie control systems[J]. J Math Anal Appl,2006,323(2):876-890.
[31]Nian X H.Delay dependent condition for absolute stability of Lurie type control systems[J].J Acta Auto Sin,1999,25:564-566(in Chinese)
[32]张景玲,包俊东.含连续分布时滞的Lurie控制系统的绝对稳定性[J].内蒙古师范大学学报,2011,40(1)：11-14.
[33]高正晖.一类含连续分布时滞的Lurie控制系统的绝对稳定性[J].广西大学学报,2007,32(4)：350-352.
[34]J.K.Hale.Theory of functional differential equations[M].New York,Springer-Verlag,1977.
[35]S.L.Campbell.Singular linear systems of differential equations with delays[J].Appl.Anal.11(1980)129-136.
[36]Zhou Zongfu,Zheng Zuxiu.Periodic Solution for Nonlinear Degenerate Systems with Delay.J.Sys.Sci.Math.Scis.,Vol.23,No.1,2003:43-50(in Chinese)
[37]Zhou Zongfu.Periodic Solution of a Class of Degenerate Differential System with Distributed Delay.Mathematica Applicata,Vol.18,No.3,2005:476-483 (in Chinese)
[38]Sun Dexian.Periodic Solution for Degenerate Differential Systems with Delay.Pure and Ap-plied Mathematics,Vol.20, No.2,2004:186-191(in Chinese)
[39]Jiang Wei.The Problem of the Periodic Solution of Singular Differential Equation with De-lay.Acta Mathematica Applicata Sinica,Vol.26,No.2,2003:180-185.
[40]陈纪修,於崇华,金路.数学分析(下)[M].北京,高等教育出版社,2000：384-426.
[41]S.D.Brierley,J.N.Chiasson,E.B.Lee,and S.H.Zak.On stability independent of delay for linear system[J]. IEEE Trans Aut,Control,1982,27(1):252-254.
[42]Feng Yifu,Zhu Xunlin and Zhang Qingling.Delay dependent stability criteria for singular time delay systems[J].Acta Automatica Sinica,2010,36(3):433-437.
[43]Xie Congyuan, Zhang Qingling.Further study of structural stability for descriptor systems[J], San Francisco California,USA,1993,3117-3120.
[44]Li Xiaoyan and Jiang Wei.On stability independence for degenerate differential systems with delay[J], J.of Math.(PRC)2004,24(5):509-512.
[45]北京大学数学系几何与代数教研室代数小组编,高等代数(第二版)[M].北京,高等教育出版社,1987：146-151.
[46]蒋威.退化中立型微分系统的常数变易公式和通解[J].应用数学学报,21(4)(1998)：562-570.
[47]蒋威.退化时滞微分系统的通解[J],数学学报,42(5)(1999)：770-780.
[48]Wang P G.Absolute stability of Lurie indirect control systems with delay time[J].Advances Simnlation,1992,29(3):43-49.
[49]徐炳吉,廖晓昕.具有控制时滞的中立型Lurie控制系统绝对稳定性[J].重庆工学院学报,2004,18(4)：11-18.
[50]徐炳吉,刘新芝.中立型Lurie控制系统的绝对稳定性判据[J].山东理工大学学报,2005,19(1)：7-16.
[51]赵小文,蒋威.变时滞退化Lurie控制系统的绝对稳定性[J].数学研究,2012,45(2)：192-197.
[52]陈纪修,於崇华,金路.数学分(第二版上册)[M].北京：高等教育出版社,2004.
[2]郑祖庥.泛函微分方程[M].合肥：安徽教育出版社,1992.
[3]李森林,温立志.泛函微分方程[M].长沙：湖南科学技术出版社,1982.
[4]谢胜利.线性定常中立型大系统的稳定性[J].控制理论与应用,1989,6(3)：7-78.
[5]S.L.Campbell.Singular Systems of Differential Equation(II),Pitman,1982.
[6]L.Dai. Singular Control Systems[M]. Berlin, New York, Springer-Verlag,1989.
[7]廖晓昕.稳定性的理论、方法和应用[M].武汉：华中科技大学出版社,2002.
[8]Jack K Hale & Sjoerd M Verduyn Lunel. Introduction to Functional Differential Equations[M]. Berlin, New York:Springer-Verlay,1992.
[9]蒋威,郑祖麻.退化中立型微分系统的常数变易公式和通解[J].应用数学学报,1998,21(4)：562-570.
[10]周宗福.一般退化时滞微分系统解的存在性及通解[J].数学研究,1998,31(4)：411-416.
[11]Hale J.Theory of functional differential equations[M].New York:Springer-Verlag,1997.
[12]郑大钟.线性系统理论[M].北京：清华大学出版社,1998.
[13]Jiang Wei. On the solvability of singular differential delay systems with variable coefficients[J]. Int.J.Dynamical Systems and Differential Equations,2008,1 (4):245-249.
[14]张海,蒋威.一般退化中立型微分系统解的存在性及通解[J].合肥工业大学学报,2007,30(5)：63-633.
[15]时宝,张德存,盖明久.微分方程理论及其应用[M].北京：国防工业出版社,2005.
[16]Jiang Wei, Zheng Zuxiu.On the degenerate differential systems with delay[J].Ann of Diff. Eqs., 1998,14(2):204-211.
[17]H.H.Rosenbrock. Structural properties of linear dynamical systems.Int.J.Contr.20(2) (1974) 191-202.
[18]Jiang Wei. Function-Controllability of Nonlinear Singular Delay Differential Control System-s[J]. Acta Mathematica Sinica,2006,49(5):1153—1162.
[19]D.G.Luenberger.Dynamical equation in descriptor form,IEEE Trans.Autom.Contr. 22(3)(1977)312-321.
[20]J.D.Cobb.Controllability,observability and duality in singular systems,IEEE Tran-s.Autom.Contr.29(1984)1076-1082.
[21]P.Kunkel,V.Mehrmann.Differential-algebraic equations.European Mathematical Society,2006.
[22]梁家荣,刘永清.广义系统的周期解,控制理论与应用.Vol.14,No.4,1997,595-598.
[23]Andras Varga,Paul,Van,Doorren.Computing the zeros of periodic descriptor systems.Systems and Control Letter.Vol.50,2003,371-381.
[24]周宗福,李蕾,王敬丰,胡秀林.一类退化中立型微分系统的周期解[J].数学物理学报,2006,26A(7)：1025-1030.
[25]Zhou Guifang.Zhang Hai,Jiang Wei.Periodic Solutions of Singular Neutral Differential Systems with Infinite Delay[J].Chin.Quart.J.of Math.2011,26(1):56-60.
[26]蒋威.退化时滞微分方程的周期解问题[J].应用数学学报,2003,26(2)：280-285.
[27]崔冬玲.含有两个时滞项的退化时滞微分方程周期解[J].生物数学学报,2010,25(2)：241-248.
[28]张作元.滞后性方程x(t)=Ax(t)+Bx(x-τ)全时滞稳定的代数判据[J].科学通报,1986：1768-1771.
[29]Lefsches S.Stability of nonlinear control systems[M].New York:Academic Press,1965.
[30]Liao X X,Yu P.Sufficient and necessary conditions for the absolute stability of time delay Lurie control systems[J]. J Math Anal Appl,2006,323(2):876-890.
[31]Nian X H.Delay dependent condition for absolute stability of Lurie type control systems[J].J Acta Auto Sin,1999,25:564-566(in Chinese)
[32]张景玲,包俊东.含连续分布时滞的Lurie控制系统的绝对稳定性[J].内蒙古师范大学学报,2011,40(1)：11-14.
[33]高正晖.一类含连续分布时滞的Lurie控制系统的绝对稳定性[J].广西大学学报,2007,32(4)：350-352.
[34]J.K.Hale.Theory of functional differential equations[M].New York,Springer-Verlag,1977.
[35]S.L.Campbell.Singular linear systems of differential equations with delays[J].Appl.Anal.11(1980)129-136.
[36]Zhou Zongfu,Zheng Zuxiu.Periodic Solution for Nonlinear Degenerate Systems with Delay.J.Sys.Sci.Math.Scis.,Vol.23,No.1,2003:43-50(in Chinese)
[37]Zhou Zongfu.Periodic Solution of a Class of Degenerate Differential System with Distributed Delay.Mathematica Applicata,Vol.18,No.3,2005:476-483 (in Chinese)
[38]Sun Dexian.Periodic Solution for Degenerate Differential Systems with Delay.Pure and Ap-plied Mathematics,Vol.20, No.2,2004:186-191(in Chinese)
[39]Jiang Wei.The Problem of the Periodic Solution of Singular Differential Equation with De-lay.Acta Mathematica Applicata Sinica,Vol.26,No.2,2003:180-185.
[40]陈纪修,於崇华,金路.数学分析(下)[M].北京,高等教育出版社,2000：384-426.
[41]S.D.Brierley,J.N.Chiasson,E.B.Lee,and S.H.Zak.On stability independent of delay for linear system[J]. IEEE Trans Aut,Control,1982,27(1):252-254.
[42]Feng Yifu,Zhu Xunlin and Zhang Qingling.Delay dependent stability criteria for singular time delay systems[J].Acta Automatica Sinica,2010,36(3):433-437.
[43]Xie Congyuan, Zhang Qingling.Further study of structural stability for descriptor systems[J], San Francisco California,USA,1993,3117-3120.
[44]Li Xiaoyan and Jiang Wei.On stability independence for degenerate differential systems with delay[J], J.of Math.(PRC)2004,24(5):509-512.
[45]北京大学数学系几何与代数教研室代数小组编,高等代数(第二版)[M].北京,高等教育出版社,1987：146-151.
[46]蒋威.退化中立型微分系统的常数变易公式和通解[J].应用数学学报,21(4)(1998)：562-570.
[47]蒋威.退化时滞微分系统的通解[J],数学学报,42(5)(1999)：770-780.
[48]Wang P G.Absolute stability of Lurie indirect control systems with delay time[J].Advances Simnlation,1992,29(3):43-49.
[49]徐炳吉,廖晓昕.具有控制时滞的中立型Lurie控制系统绝对稳定性[J].重庆工学院学报,2004,18(4)：11-18.
[50]徐炳吉,刘新芝.中立型Lurie控制系统的绝对稳定性判据[J].山东理工大学学报,2005,19(1)：7-16.
[51]赵小文,蒋威.变时滞退化Lurie控制系统的绝对稳定性[J].数学研究,2012,45(2)：192-197.
[52]陈纪修,於崇华,金路.数学分(第二版上册)[M].北京：高等教育出版社,2004.